2020 AIME II Problems/Problem 1
Find the number of ordered pairs of positive integers such that .
In this problem, we want to find the number of ordered pairs such that . Let . Therefore, we want two numbers, and , such that their product is and is a perfect square. Note that there is exactly one valid for a unique , which is . This reduces the problem to finding the number of unique perfect square factors of .
Therefore, the answer is
Solution 2 (Official MAA)
Because , if , there must be nonnegative integers , , , and such that and . Then and The first equation has solutions corresponding to , and the second equation has solutions corresponding to . Therefore there are a total of ordered pairs such that .
~ North America Math Contest Go Go Go
Purple Comet Math Meet April 2020
Notice, that this was the exact same problem (with different wording of course) as Purple Comet HS problem 3 and remembering the answer, put .
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